Mixer sequence design for N-path filters

ABSTRACT

A bandpass filter includes a plurality of parallel paths, each receiving the input signal to the bandpass filter. Each path includes a first mixer, a low-pass filter, and a second mixer. The first mixer in each path is coupled to receive the input signal and mixes the input signal with a periodic mixer sequence having a period that is divided into a plurality of time slots. The mixer value is constant during each time slot. The low-pass filter in each path is operable to filter an output of the associated first mixer. The second mixer in each path is coupled to receive an output of the associated low-pass filter and mixes said filter output with a periodic mixer sequence having a period that is divided into a plurality of time slots, wherein again the mixer value is constant during each time slot. A summer sums the outputs of the second mixers of each of the paths to generate an output of the bandpass filter.

BACKGROUND

Bandpass filters are used in many applications including radio frequencyreceiver paths and bandpass delta-sigma analog-to-digital converters.Typical bandpass filter implementations require inductors, and it isdifficult to implement high quality, appropriately sized, inductors incomplementary metal oxide semiconductor (CMOS) processes. Other optionsexist for implementing bandpass filters, but they tend to have issueswith performance and/or power, or require alternative processtechnologies.

N-path filters are a practical method for implementing high-Q bandpassfilters in modern CMOS processes without inductors using a combinationof mixers and low-pass filters. The Q factor is the ratio of the centerfrequency of the filter to the pass band bandwidth. The basic structureof an N-path filter is multiple paths, each path composed of a mixer,filter and mixer, summed together to form the filter output. With trendsin process scaling leading to higher switching frequencies, N-pathfilters are a viable option for integrated bandpass filter designs withcenter frequencies of interest in current communication standards. Asthe center frequency of the filter is decoupled from the bandwidth ofthe filter, high Q values are achievable. The mixer sequences have ahigh impact on the performance of the N-path filter. Thereforeoptimizing practically realizable mixer sequences allows optimization ofthe N-path filter.

SUMMARY

A bandpass filter in accordance with one illustrative embodiment of thepresent invention includes a plurality of parallel paths, each receivingthe input signal to the bandpass filter. Each path includes a firstmixer, a low-pass filter, and a second mixer. The first mixer in eachpath is coupled to receive the input signal and mixes the input signalwith a periodic mixer sequence having a period that is divided into aplurality of time slots. The mixer value is constant during each timeslot. The low-pass filter in each path is operable to filter an outputof the associated first mixer. The second mixer in each path is coupledto receive an output of the associated low-pass filter and mixes saidfilter output with a periodic mixer sequence having a period that isdivided into a plurality of time slots, wherein again the mixer value isconstant during each time slot. A summer sums the outputs of the secondmixers of each of the paths to generate an output of the bandpassfilter.

Another embodiment of the invention is directed to a bandpass filterthat includes a first path and a second path. The first path includes afirst mixer, a first low-pass filter, and a second mixer. The firstmixer mixes an input signal with a sampled cosine signal to produce afirst mixed signal. The first low-pass filter is operable to low-passfilter the first mixed signal to produce a first filtered signal. Thesecond mixer mixes the first filtered signal with the sampled cosinesignal to produce a second mixed signal. The second path includes athird mixer, a second low-pass filter, and a fourth mixer. The thirdmixer mixes the input signal with a sampled sine signal to produce athird mixed signal. The second low-pass filter low-pass filters thethird mixed signal to produce a second filtered signal. The fourth mixermixes the second filtered signal with the sampled sine signal to producea fourth mixed signal. A summer sums the second mixed signal and thefourth mixed signal to produce an output of the bandpass filter.

Another embodiment of the invention is directed to a bandpass filterthat has N parallel paths, each path arranged to receive the inputsignal to the bandpass filter. Each path includes a first mixer, alow-pass filter, and a second mixer. The first mixer in each path iscoupled to receive the input signal and mixes the input signal with aperiodic two-level mixer sequence having a period of length T that isdivided into M time slots, where M is an integer greater than one. Themixer value is constant during each time slot. Assume M is an evennumber. The number of paths N is greater than or equal to M/2. Only thefirst M/2 paths are active, where the mixer sequence p of the firstmixer of the n^(th) path (1≦n≦M/2) is chosen to satisfy

${p^{(n)}(t)} = {p^{(1)}\left( {t - \frac{\left( {n - 1} \right)T}{M}} \right)}$and

${p^{(1)}\left( {t + \frac{T}{2}} \right)} = {- {{p^{(1)}(t)}.}}$The low-pass filter in each path is operable to filter an output of theassociated first mixer. The second mixer in each path is coupled toreceive an output of the associated low-pass filter and mix said filteroutput with a periodic two-level mixer sequence having a period oflength T that is divided into M time slots. The mixer value is constantduring each time slot. The mixer sequence q of the second mixer of then^(th) path (1≦n≦M/2) is chosen to satisfy

${q^{(n)}(t)} = {{q^{(1)}\left( {t - \frac{\left( {n - 1} \right)T}{M}} \right)}.}$A summer sums the outputs of the second mixers of each of the paths togenerate an output of the bandpass filter.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram of an N-path filter.

FIG. 2 is a schematic diagram of a bandpass filter comprising an N-pathfilter.

FIG. 3 is a flow chart representing a method of bandpass filtering aninput signal using an N-path filter that employs mixers that haveperiodic staircase mixer sequences.

FIG. 4 is a flow chart representing a method of bandpass filtering aninput signal using an N-path filter that employs mixers that useperiodic sampled sine and cosine mixer sequences.

FIG. 5 is a flow chart representing a method of bandpass filtering aninput signal using an N-path filter that employs mixers that usetwo-level periodic mixer sequences and N≧M/2, where M is the number ofslots in each period of the mixer sequences.

FIG. 6 is a flow chart representing a method of bandpass filtering aninput signal using an N-path filter that employs mixers that usetwo-level periodic mixer sequences and N<M/2, where M is the number ofslots in each period of the mixer sequences.

DETAILED DESCRIPTION

The present invention is directed generally to an N-path bandpass filterhaving mixer sequences that are constrained to a staircase sequence andto a two-level sequence.

FIG. 1 is a schematic diagram of an N-path bandpass filter 100. Thefirst path includes a mixer 105, a low-pass filter 120, and a secondmixer 135. The second path includes a mixer 110, a low-pass filter 125,and a second mixer 140. The N^(th) path includes a mixer 115, a low-passfilter 130, and a second mixer 145. The first mixers 105, 110, 115 ineach path have a mixer signal p^(n)(t), and the second mixers 135, 140,145 in each path have a mixer signal q^((n))(t). The mixers have aperiod T which determines the center frequency Ω₀=2π/T of the bandpassfilter. The products of each path are summed together by summer 150 toform the filter output. The mixers transform the low-pass filter shapeto a pass band around Ω₀, where the double-sided bandwidth of thelow-pass filter is the same as the bandwidth of the bandpass filter. Inan illustrative embodiment, the N-path filter 100 is implemented incomplementary metal-oxide-semiconductor (CMOS) integrated circuit(s).But other technologies, including other semiconductor technologies, canbe used as well.

It can be shown mathematically that the N-path filter 100 transforms alow-pass-filter into a bandpass filter. Defining X(jΩ) as the input andY(jΩ) as the output spectrum of the N-path filter, and denoting H(jΩ) asthe low-pass filter, the output of the N-path filter can be representedas:

$\begin{matrix}{{{Y\left( {j\;\Omega} \right)} = {\sum\limits_{r = {- \infty}}^{\infty}{{H\left( {j\left( {\Omega - {r\;\Omega_{0}}} \right)} \right)} \cdot {Y_{r}\left( {j\;\Omega} \right)}}}},} & (1)\end{matrix}$in which the input and mixer related terms are

$\begin{matrix}{{{Y_{r}\left( {j\;\Omega} \right)} = {\sum\limits_{m = {- \infty}}^{\infty}{{X\left( {j\left( {\Omega + {\left( {m - r} \right)\;\Omega_{0}}} \right)} \right)} \cdot {\alpha\left( {m,r} \right)}}}},} & (2)\end{matrix}$and

$\begin{matrix}{{{\alpha\left( {m,r} \right)} = {{\sum\limits_{n = 1}^{N}{{\hat{p}}_{- m}^{(n)} \cdot {\hat{q}}_{r}^{(n)}}} = {\sum\limits_{n = 1}^{N}{\left( {\hat{p}}_{m}^{(n)} \right)^{*} \cdot {\hat{q}}_{r}^{(n)}}}}},} & (3)\end{matrix}$where {circumflex over (p)}_(m) ^((n)) and {circumflex over (q)}_(m)^((n)) are the m^(th) Fourier series coefficients of p^(n)(t) andq^(n)(t), respectively. Typically, the bandwidth BW<<Ω₀ and Eq. (1)implies that the output only has significant power in frequencies ±BW/2around the harmonics of Ω₀.

For simplicity, it can be assumed that the power is flat in frequencies±BW/2 around a harmonic, so only the midpoint of each band (i.e.,Y(j·lΩ₀)) is considered. When Ω=lΩ₀ in Eq. (1), only the r=l termremains. Since H(j0) is the same scale factor for all harmonics, letH(j0)=1 to obtain

$\begin{matrix}{{Y\left( {{j \cdot l}\;\Omega_{0}} \right)} \approx {\sum\limits_{m = {- \infty}}^{\infty}{{X\left( {{j \cdot m}\;\Omega_{0}} \right)} \cdot {{\alpha\left( {m,l} \right)}.}}}} & (4)\end{matrix}$a(m,l) can be viewed as the transfer coefficient from the m^(th)harmonic in the input to the l^(th) harmonic in the output.

It can further be assumed for simplicity's sake that the stationaryinput signals at different harmonics are uncorrelated and both X(j·mΩ₀)and X²(j·mΩ₀) have zero mean. Denoting E{·} as the average operator, theoutput signal power spectral density (PSD) at the l^(th) harmonic is:

$\begin{matrix}{{E\left\{ {{Y\left( {{j \cdot l}\;\Omega_{0}} \right)}}^{2} \right\}} \approx {\sum\limits_{m = {- \infty}}^{\infty}{E{\left\{ {{X\left( {{j \cdot m}\;\Omega_{0}} \right)}}^{2} \right\} \cdot {{{\alpha\left( {m,l} \right)}}^{2}.}}}}} & (5)\end{matrix}$

The in-band output corresponds to l=1 and has two components: thein-band signal and the folded harmonic. The desired in-band signal,which is the output of a traditional bandpass filter, corresponds to theterm l=m=1 and has average power

$\begin{matrix}{P_{signal} = {E{\left\{ {{X\left( {j\;\Omega_{0}} \right)}}^{2} \right\} \cdot {{{\alpha\left( {1,1} \right)}}^{2}.}}}} & (6)\end{matrix}$The unwanted folded harmonics can be viewed as interference to thein-band signal. They correspond to terms with l=1, m≠1 in (4) and haveaverage total power

$\begin{matrix}{P_{folded} = {\sum\limits_{m \neq 1}{E{\left\{ {{X\left( {{j \cdot m}\;\Omega_{0}} \right)}}^{2} \right\} \cdot {{{\alpha\left( {m,1} \right)}}^{2}.}}}}} & (7)\end{matrix}$In addition to the in-band output, the N-path filter typically hasout-of-pass-band outputs around the harmonics of Ω₀. For the l^(th)harmonic (l≠±1), the average out-of-band power is

$\begin{matrix}{{P_{out}(l)} = {\sum\limits_{m = {- \infty}}^{\infty}{E{\left\{ {{X\left( {{j \cdot m}\;\Omega_{0}} \right)}}^{2} \right\} \cdot {{{\alpha\left( {m,l} \right)}}^{2}.}}}}} & (8)\end{matrix}$

The above analysis shows that the N-path filter transforms a low-passfilter to a bandpass filter with two nonidealities: in-band harmonicfolding and out-of-band signal residue. FIG. 2 shows a bandpass filterarrangement 200 that reduces these two nonideal effects. The bandpassfilter 200 employs a loose pre-low-pass filter 210 and a loosepost-low-pass filter 230 around the N-path filter 220, such as theN-path filter 100 of FIG. 1. With cutoff frequencies a little above Ω₀,the pre-low-pass filter 210 attenuates signals at high harmonics toavoid folding onto the in-band signal and the post-low-pass filter 230removes residual out-of-band signal power.

Alternatively, as a(m,l) in Eqs. (6-8) depends on the Fouriercoefficients of the mixer signals, it's possible to design mixer signalsthat reduce the in-band harmonic folding and the out-of-band signalresidue such that the requirements on the pre and post low-pass filters210 and 230 are reduced or eliminated. The design of the mixer sequencesp^((n))(t) of the mixers 105, 110, 115, and the mixer sequencesq^((n))(t) of the mixers 135, 140, 145 for this purpose is exploredbelow.

In an illustrative embodiment of the present invention, the mixersequences p^((n))(t) of the mixers 105, 110, 115, and the mixersequences q^((n))(t) of the mixers 135, 140, 145 have periodic staircasesequences. Each period is split into M equal time slots and each mixervalue is constant within each time slot. FIG. 3 is a flow-chartrepresenting a method of bandpass filtering an input signal using anN-path filter that employs mixers that have periodic staircase mixersequences. At block 300, an input signal, such as signal x(t) shown inFIG. 1, is received. At block 310, the input signal is provided to eachof a plurality of parallel paths. Each path includes a first mixer, alow-pass filter, and a second mixer. At block 320, in each path, thefirst mixer 105, 100, 115 mixes the input signal with a periodic mixersequence having a period that is divided into a plurality of time slots.The mixer value is constant during each time slot. At block 330, thelow-pass filter 120, 125, 130 in each path low-pass filters the outputof the associated first mixer 105, 110, 115. At block 340, the secondmixer 135, 140, 145 in each path receives the output of the associatedlow-pass filter and mixes said filter output with a periodic mixersequence having a period that is divided into a plurality of time slots,wherein the mixer value is constant during each time slot. At block 350,the summer 150 sums the outputs of the second mixers 135, 140, 145 ofeach of the paths to generate an output of the bandpass filter.

For purposes of the present invention, two criteria are considered inevaluating N-path filters: in-band signal-to-noise ratio (SNR) andout-of-band harmonic power ratio. If the folded harmonics are consideredas in-band noise, then the goal is to maximize the in-bandSNR=P_(signal)/P_(folded). Regarding the out-of-band harmonic powerratio, for l≠±1, the goal is to minimizeR_(out)(l)=P_(out)(l)/P_(signal).

The following lemma and corollary, which put an upper limit on theachievable in-band SNR and a lower limit on the out-of-band harmonicpower ratio, are used in design of the mixer sequences p^((n))(t) of themixers 105, 110, 115, and the mixer sequences q^((n))(t) of the mixers135, 140, 145. The lemma states that, for M-slot staircase mixersequences, the harmonic power ratio for the lth harmonic is lowerbounded by

$\begin{matrix}{{R_{out}(l)} = {{{P_{out}(l)}/P_{signal}} \geq \left\{ {\begin{matrix}{{\sum\limits_{b = {- \infty}}^{\infty}\frac{E\left\{ {{X\left( {{j\left( {{bM} + 1} \right)}\Omega_{0}} \right)}}^{2} \right\}}{{\left( {l\left( {{bM} + 1} \right)} \right)^{2} \cdot E}\left\{ {{X\left( {j\;\Omega_{0}} \right)}}^{2} \right\}}},{{{for}\mspace{14mu} l} = {{cM} + 1}}} \\{{\sum\limits_{b = {- \infty}}^{\infty}\frac{E\left\{ {{X\left( {{j\left( {{bM} - 1} \right)}\Omega_{0}} \right)}}^{2} \right\}}{{\left( {l\left( {{bM} - 1} \right)} \right)^{2} \cdot E}\left\{ {{X\left( {j\;\Omega_{0}} \right)}}^{2} \right\}}},{{{for}\mspace{14mu} l} = {{cM} - 1}}} \\{0,{otherwise}}\end{matrix},} \right.}} & (9)\end{matrix}$and the lower bound is achieved ifa(m,l)=0, for 0≦m<M, 0≦l<M,except for (m,l)=(1,1) or (M−1,M−1).  (10)

The ratio a(bM+1, cM+1)/a(1,1) is independent of the input, where a(1,1)controls P_(signal) and a(bM+1,cM+1) controls the power contributed fromthe (bM+1)^(th) input harmonic to P_(out)(cM+1). The bound is achievedwhen no other input harmonics contribute to the (cM+1)^(th) outputharmonic. The case l=cM−1 is due to spectrum symmetry. Setting l=1 inthe lemma leads to the corollary: with M-slot staircase mixer sequences,the in-band signal-to-noise ratio is upper bounded by

$\begin{matrix}{{{SNR} \leq \frac{E\left\{ {{X\left( {j\;\Omega_{0}} \right)}}^{2} \right\}}{\sum\limits_{b\bumpeq\not{}0}\frac{E\left\{ {{X\left( {{j\left( {{bM} + 1} \right)}\Omega_{0}} \right)}}^{2} \right\}}{\left( {{bM} + 1} \right)^{2}}}},} & (11)\end{matrix}$and the upper bound is achieved ifa(m,1)=0, for m≠1,0≦m<M.  (12)Staircase Sequences

In one illustrative embodiment of the M-slot staircase mixer sequencesof the present invention, the signal amplitude can vary from time slotto time slot and from path to path, and there is no constraint on therelationship between the mixer signals on different paths or the numberof paths N. In this scenario, a two-path sampled quadrature filterachieves both the optimal in band signal-to-noise ratio and the optimalharmonic power ratio at each harmonic frequency, among all N-pathfilters with M-slot staircase mixer sequences. In the period [0,T], themixer sequences in the two-path sampled quadrature filter have valuesp ⁽¹⁾(t)=q ⁽¹⁾(t)=cos(2πm/M)p ⁽²⁾(t)=q ⁽²⁾(t)=sin(2πm/M)  (13)where (mT/M)≦t<((m+1)T/M) and 0≦m≦M−1.

Thus a bandpass filter according to this embodiment comprises just twopaths. Referring again to FIG. 1, such a two-path filter would include afirst path comprising first mixer 105, low-pass filter 120, and secondmixer 135, and a second path comprising first mixer 110, low-pass filter125, and second mixer 140. Both mixers 105 and 135 in the first pathutilize a mixer sequence comprising a sampled cosine signal. Both mixers110 and 140 in the second path utilize a mixer sequence comprising asampled sine signal.

To prove that this two-path scheme employing sampled sine and cosinemixer sequences is the optimal N-path filter that uses periodicstaircase mixer sequences, it is necessary only to test whether thesequences of (13) satisfy (10) and (12). Note that for 0≦m≦M−1, the onlynonzero terms of Fourier series coefficients of p^((n))(t) andq^((n))(t) in (13) are {circumflex over (p)}₁ ^((n)), {circumflex over(p)}_(M−1) ^((n)), {circumflex over (q)}₁ ^((n)), and {circumflex over(q)}_(M−1) ^((n)). Thus, for 0≦m, l≦M−1, the only possible nonzero termsof a(m,l) are a(1,1), a(1,M−1), a(M−1,1) and a(M−1,M−1). Directcalculation can verify that a(1,M−1)=a(M−1,1)=0. As such, (10) issatisfied and the minimum harmonic power ratio at each harmonic isachieved.

Constraint (12) for the optimal in-band signal-to-noise ratio is aspecial case of (10) with l=1 which is already satisfied. Thus, thesampled sine and cosine mixer signals of (13) achieve both maximumin-band signal-to-noise ratio and minimum harmonic power ratio at eachharmonic.

FIG. 4 is a flow-chart representing a method of bandpass filtering aninput signal using an N-path filter that employs mixers that use sampledsine and cosine mixer sequences. At block 400, an input signal, such assignal x(t) shown in FIG. 1, is received. The input signal is providedto a first path and a second path. The first path, represented by blocks410, 420 and 430, includes a first mixer, a low-pass filter, and asecond mixer. At block 410, the first mixer mixes the input signal witha sampled cosine signal to produce a mixed signal. At block 420, thelow-pass filter low-pass filters the mixed signal to produce a low-passfiltered signal. At block 430, the second mixer mixes the low-passfiltered signal with the sampled cosine signal to produce a second mixedsignal. The second path, represented by blocks 440, 450 and 460, alsoincludes a mixer, a low-pass filter, and a second mixer. At block 440,the second path's first mixer mixes the input signal with a sampled sinesignal to produce a mixed signal. At block 450, the low-pass filterfilters the mixed signal to produce a low-pass filtered signal. At block460, the second mixer of the second path mixes the low-pass filteredsignal with the sampled sine signal to produce a second mixed signal. Atblock 470, a summer sums the mixed signal received from the first andsecond paths to produce an output of the bandpass filter.

It is noted that, if the mixers of the N-path filter such as that shownin FIG. 1 are constrained to staircase sequences, then adding more thantwo paths does not provide a gain for in-band signal-to noise-ratio orout-of-band signal rejection. Also, the optimal mixer sequences areindependent of the power spectral density (PSD) of the input signal andthe location of any blockers. Improving the in-band signal-to-noiseratio and out-of-band signal rejection requires an increase of thenumber of time slots M in one period, i.e., the system has to run at ahigher clock frequency.

Two-level Sequences

In another illustrative embodiment of the M-slot staircase mixersequences of the present invention, the mixer sequences are furtherconstrained to taking on one of only two values in each of the M slots.Thus:p ^((n))(t)ε{1,−1}, q ^((n))(t)ε{A _(n) ,−A _(n)},  (14)where A_(n) is a constant gain for the n^(th) path. Limiting the mixersto two levels makes them easier to implement in analog.

In the following design, it is assumed for the sake of simplicity ofexplanation that M is an even number. If M is odd, the results aresimilar as will be explicitly described subsequently. For an even M, thecases of N≧M/2 and N<M/2 paths are separately considered. Additionally,only the in-band signal-to-noise ratio criterion is considered in theanalysis of the two-level sequences.

Two-level Sequences with N≧M/2 Paths

Among N-path filters whose mixer sequences have M slots per period andare constrained to taking on one of only two values in each of the Mslots, the M/2-path filter with the following class of mixer sequencesachieves the optimal in-band signal-to-noise ratio:p ^((n))(t)=p ⁽¹⁾(t−((n−1)T/M)),  (15)q ^((n))(t)=q ⁽¹⁾(t−((n−1)T/M)),  (16)p ⁽¹⁾(t+(T/2))=−p ⁽¹⁾(t).  (17)

The antisymmetric condition of Eq. (17) indicates that p^((n))(t) has noeven harmonics. Thus {circumflex over (p)}_(2m) ^((n))=0 and a(2m,1)=0.The delay relationships in Eqs. (15) and (16) result in a phase factorin the Fourier series coefficients and it can be verified thata(2m+1,1)=0 for 1≦m<M/2. Thus, (12) is satisfied and the optimal in-bandSNR is achieved.

In particular, the half-plus half-minus (HPHM) sequences

$\begin{matrix}{{p^{(1)}(t)} = \left\{ {\begin{matrix}{1,} & {{0 \leq t < \left( {T/2} \right)},} \\{{- 1},} & {\left( {T/2} \right) \leq t < T}\end{matrix},} \right.} & (18)\end{matrix}$p ^((n))(t)=p ⁽¹⁾(t−((n−1)T/M)),  (19)q ^((n))(t)=p ^((n))(t)  (20)satisfy the constraints of (15-17) and have optimal in-band SNR. Thesesequences are particularly implementation friendly as there are only twolevel changes in one period in each path, and mixer sequences inconsecutive paths have a delay of one slot.

FIG. 5 is a flow-chart representing a method of bandpass filtering aninput signal using an N-path filter that employs mixers that usetwo-level mixer sequences. At block 500, an input signal, such as signalx(t) shown in FIG. 1, is received. At block 510, the input signal isprovided to each of a plurality of parallel paths. Each path includes afirst mixer, a low-pass filter, and a second mixer. At block 520, ineach path, the first mixer 105, 100, 115 mixes the input signal with aperiodic two-level mixer sequence having a period that is divided into Mtime slots, wherein N≧M/2. The mixer value is constant during each timeslot and the mixer sequence p satisfies Eq. (15) and Eq. (17). At block530, the low-pass filter 120, 125, 130 in each path low-pass filters theoutput of the associated first mixer 105, 110, 115. At block 540, thesecond mixer 135, 140, 145 in each path receives the output of theassociated low-pass filter and mixes said filter output with a periodictwo-level mixer sequence having a period that is divided into M timeslots, wherein the mixer value is constant during each time slot andwherein N≧M/2. The mixer sequence q satisfies Eq. (16). At block 550,the summer 150 sums the outputs of the second mixers 135, 140, 145 ofeach of the paths to generate an output of the bandpass filter.

It is noted that additional paths beyond M/2 do not provide a gain forin-band signal-to noise ratio. However, out-of-band signal rejectioncould potentially be improved. Additionally, the optimal sequences areindependent of the input signal power spectral density.

Two-level Sequences with N<M/2 Paths

N-path filters whose M-slot mixer sequences are constrained to taking onone of only two values in each of the M slots, and where the number ofpaths N is restricted to N<M/2, are now considered. Only the in-bandsignal-to-noise ratio criterion is considered in the analysis of thesetwo-level sequences. In contrast to the previous results, an inputsignal independent mixer sequence is not obtained under theseconstraints. Instead, a heuristic optimization algorithm is proposed.

Let v_(k) ^((n)) and w_(k) ^((n)) represent the values in the kth timeslot (1≦k≦M) of the mixer sequences p^((n))(t) and q^((n))(t),respectively. The in band signal-to-noise ratio has the form of

$\begin{matrix}{{SNR} = {\frac{E{\left\{ {{X\left( {j\;\Omega_{0}} \right)}}^{2} \right\} \cdot {{\alpha\left( {1,1} \right)}}^{2}}}{\sum\limits_{m\bumpeq\not{}1}{E{\left\{ {{X\left( {{j \cdot m}\;\Omega_{0}} \right)}}^{2} \right\} \cdot {{\alpha\left( {m,1} \right)}}^{2}}}}.}} & (21)\end{matrix}$Since the Fourier coefficients {circumflex over (p)}_(m) ^((n)) and{circumflex over (q)}₁ ^((n)) are linear in the values of v_(k) ^((n))and w_(k) ^((n)), respectively, a(m,1) is a bilinear form with respectto v_(k) ^((n)) and w_(k) ^((n)). Therefore, the powers of the signaland folded harmonic are both quadratic forms with respect to eitherv_(k) ^((n)) and w_(k) ^((n)). The total order of 4 is a challenge foroptimizing the signal-to-noise ratio per Eq. (21).

To reduce the order of the objective function, an iterative two-partheuristic algorithm is used. The first part (i.e., part 1) optimizesover v_(k) ^((n)) with w_(k) ^((n)) held constant. The second part(i.e., part 2) optimizes over w_(k) ^((n)) with v_(k) ^((n)) heldconstant.

The optimization problem of part 1 can be written as

$\begin{matrix}{{{\max\limits_{v \in {\{{{- 1},1}\rbrack}^{MN}}{SNR}} = \frac{v^{T}{Sv}}{v^{T}{Nv}}},} & (22)\end{matrix}$where S and N are positive semidefinite matrices dependent on N, M,w_(k) ^((n)), and the input signal power spectral densityE{|X(j·mΩ₀)|²}. If Nv≠0 for all “binary” v vectors, it can be shown thatthe optimal objective function of Eq. (22) has the value of λ if andonly if the following problem

$\begin{matrix}{\max\limits_{v \in {\{{{- 1},1}\rbrack}^{MN}}\left( {{v^{T}{Sv}} - {{\lambda \cdot v^{T}}{Nv}}} \right)} & (23)\end{matrix}$has a maximum of 0. Problem (23) is an unconstrained binary quadraticprogramming problem and can be approximately solved by greedy localsearch. The solution of (23) never decreases signal-to-noise ratio,which typically leads to convergence. Thus, the optimization problem inpart 1 can be heuristically solved by an iterative algorithm.

FIG. 6 is a flow chart representing the first part of the iterativeheuristic algorithm according to an illustrative embodiment of thepresent invention. At block 600, the vectors v and w are initializedwith the half-plus half-minus sequences of (18)-(20). At block 610, thecurrent λ is computed as λ=(v^(T)Sv)/(v^(T)Nv). At block 620, problem(23) is solved with the current λ. At block 630, the vector v is updatedwith the solution to problem (23). At decision block 640, it isdetermined if the optimal cost is zero. If the optimal cost is zero,part 1 of the heuristic algorithm terminates, as shown at block 660. Ifthe optimal cost is not zero, it is determined whether the limit oniteration steps has been reached at decision block 650. If the limit oniteration steps is reached, then part 1 of the heuristic algorithmterminates at block 660. If the limit on iteration steps is not reached,the algorithm is repeated starting at block 610.

For the second part of the heuristic algorithm, since the in-bandsignal-to-noise ratio involves only {circumflex over (q)}₁ ^((n)), theoptimization is performed on [{circumflex over (q)}₁ ⁽¹⁾, . . . ,{circumflex over (q)}₁ ^((N))]ε

^(N). Equation (21) is the ratio of semidefinite quadratic forms withrespect to {circumflex over (q)}₁ ^((n)), hence, its solution isavailable in closed form. After solving for the optimal {circumflex over(q)}₁ ^((n)), its phase is quantized into delays which are multiples ofT/M.

The two parts of the algorithm may take multiple iterations to determinea heuristic-based optimal solution for the two-level mixer sequences ineach path. There are potentially local maximums and no guarantees ofglobal optimality are provided. In contrast to the staircase mixersequence or two-level sequence with N≧M/2 paths, the optimal sequence inthe case N<M/2 paths depends on the input signal and the number ofpaths.

In the above analysis of the two-level sequences, the number of timeslots per period M is assumed an even number. If M is odd, thederivation is very similar and thus omitted. However, there aredifferences in certain conclusions, which are stated as follows. For anodd M, the cases of N≧M and N<M paths are separately considered. If N≧M,then the M-path filter with sequences satisfying Eq. (15) and (16)achieves the maximum in-band signal-to-noise ratio among all N-pathfilters with M-slot two-level mixer sequences. This optimum isindependent of the input signal power spectral density; furthermore,additional paths beyond M does not improve in-band signal-to-noiseratio. In contrast, if N<M, the optimal sequences can be obtained by theheuristic optimization algorithm introduced previously. In this case,the optimal sequences depend on the input signal and the number ofpaths.

Having thus described circuits and methods for implementing an N-pathbandpass filter by reference to certain of its preferred embodiments, itis noted that the embodiments disclosed are illustrative rather thanlimiting in nature and that a wide range of variations, modifications,changes, and substitutions are contemplated in the foregoing disclosure.For example, in an illustrative embodiment of the invention, the N-pathfilter 100 is implemented in complementary metal-oxide-semiconductor(CMOS) integrated circuit(s), but other technologies, including othersemiconductor technologies, can be used as well. Furthermore, in someinstances, some features of the present invention may be employedwithout a corresponding use of the other features. Accordingly, it isappropriate that the appended claims be construed broadly and in amanner consistent with the broad inventive concepts disclosed herein.

What is claimed is:
 1. A bandpass filter comprising: a plurality ofparallel paths, each path arranged to receive an input signal to thebandpass filter, each path comprising: a first mixer coupled to receivethe input signal and operable to mix the input signal with a periodictwo-level mixer sequence having a period that is divided into aplurality of time slots, wherein the mixer value is constant during eachtime slot; a low-pass filter operable to filter an output of theassociated first mixer; and a second mixer coupled to receive an outputof the associated low-pass filter and operable to mix said filter outputwith a periodic two-level mixer sequence having a period that is dividedinto a plurality of time slots, wherein the mixer value is constantduring each time slot; and a summer operable to sum the outputs of thesecond mixers of each of the paths to generate an output of the bandpassfilter; wherein the mixer sequences of the first and second mixers ofeach path have a period that is divided into M time slots, where M is aninteger greater than one, and wherein the number of paths N≧M/2.
 2. Thebandpass filter of claim 1 wherein the mixer sequence p of the firstmixer of the n^(th) path (1≦n≦M/2) satisfies${{p^{(n)}(t)} = {p^{(1)}\left( {t - \frac{\left( {n - 1} \right)T}{M}} \right)}},$wherein the mixer sequence q of the second mixer of the n^(th) path(1≦n≦M/2) satisfies${{q^{(n)}(t)} = {q^{(1)}\left( {t - \frac{\left( {n - 1} \right)T}{M}} \right)}},$and wherein${p^{(1)}\left( {t + \frac{T}{2}} \right)} = {- {{p^{(1)}(t)}.}}$
 3. Thebandpass filter of claim 2 wherein${p^{(1)}(t)} = \left\{ {\begin{matrix}{1,} & {{0 \leq t < \left( {T/2} \right)},} \\{{- 1},} & {\left( {T/2} \right) \leq t < T}\end{matrix},} \right.$ andp ^((n))(t)=p ⁽¹⁾(t−((n−1)T/M)), and whereinq ^((n))(t)=p ^((n))(t), where 1≦n≦M/2.
 4. The bandpass filter of claim1 wherein the bandpass filter comprises a complementary metal-oxidesemiconductor (CMOS) integrated circuit.